Conventionally, in order to manufacture a metal polygonal can, e.g., in the case of manufacturing a two-piece can, there has been generally known a method in which a blank is subjected to drawing and ironing to form a bottomed cylindrical base-can and the cylindrical base-can is then formed into a bottomed polygonal can by a square die and a square punch (for example, Patent Literature 1). In addition, as another method, there has been known an impact extrusion molding method in which a bottomed polygonal can is formed from a slug made of a soft aluminum alloy by a punch and a die in a single step. In the case of manufacturing a three-piece can, there has been known a bulge forming method in which a bottomless cylindrical base-forming is internally expanded by water pressure to form a can body along the shape of an outer mold or a method in which a seam-bonded bottomless cylindrical body is pressed inward by an outer die having a processing surface with a mountain shape in cross section to form a body wall part and some parts of the cylindrical body that are expected to form corner parts are pressed outward by an inner die having a processing surface with an arc shape in cross section at its tip end to form the corner parts (Patent Literature 2). The latter method does not assume the forming of a bottomed cylindrical can into a bottomed polygonal can, and thus a bottomed polygonal can cannot be formed by this method.